Time-Reversal Symmetry Bounds on the Electromagnetic Response of Asymmetric Structures

Asymmetric structures support different field distributions and electromagnetic responses when excited from different directions. Here we show that time-reversal symmetry imposes fundamental constraints on their overall response, beyond those dictated by reciprocity. For two-port devices, the asymmetry in field distribution for opposite excitations is shown to be fundamentally bounded by the reflection at the ports, and the fields are identical everywhere in space in the case of full transmission. In multiport and open scenarios, these bounds have implications on radiation and scattering at different ports and towards different directions. Beyond their theoretical significance, these results provide relevant insights into the operation of nonlinear isolators, metasurfaces, and other nanophotonic devices.

Figure 1

(a) Two-port asymmetric structure without loss. The effect of time reversal is presented at the bottom. (b) Multiport asymmetric structure. (c) Asymmetric object in free space illuminated by a plane wave propagating along the $z$ axis.

Figure 2

Numerical validation of the time-reversal-symmetry bound for two-port structures. (a) Multimodal cavity inside a parallel-plate waveguide. All the walls are PEC and the material of the cavity has ${ϵ}_r=12$. The cavity parameters are $h_1=0.9a$, $h_2=0.7a$, $w_1=a$. The operation frequency is $0.35c/a$ and the electric field is polarized along the $y$ axis with amplitude $1\mathrm{V}/\mathrm{m}$. (b) Same as in (a), but with an additional matching layer at the right-hand side of the cavity, which consists of metallic patches on both sides of a dielectric substrate with permittivity ${ϵ}_r=12$. The parameters are $d=1.449a$, $w_2=0.1a$, and $h_3=0.81a$. The color bar scale is different than in (a) in order to properly show the details of the fields, which are stronger in this case. (c) Field ratio versus reflectivity for excitation from opposite sides for the structure in (b) at 25 equally distributed points inside the cavity and 101 equally distributed frequencies over a band centered at frequency $0.35c/a$ and with width $0.03c/a$. The red lines correspond to the upper and lower bounds in Eq. (1) for the field intensity ratio. All simulations have been performed with comsol multiphysics.

Figure 3

Multiport and multimodal asymmetric structure. (a) The structure is infinitely extended along the $z$ direction, is filled with air, and has PMC walls. The cavity vertices are $P_1=(0,0)\mathrm{m}$, $P_2=(-0.1,0.28)\mathrm{m}$, $P_3=(0.29,0.51)\mathrm{m}$, $P_4=(0.44,0.32)\mathrm{m}$, and $P_5=(0.3,0)\mathrm{m}$. The width of the external channels is 0.03 m. (b) $S$ parameters for excitation from port 1. (c) Radiation efficiency at different ports for a dipole source located at six different evenly distributed points inside the cavity, including the center of mass $O={\sum }_{k=1}^5{P}_k/5$ and the midpoints between $O$ and the cavity vertices. The shaded regions correspond to the radiation efficiencies allowed by time-reversal symmetry. The simulations were carried out through comsol multiphysics for $z$-polarized incident waves with amplitude $1\mathrm{V}/\mathrm{m}$ and uniform profile across the waveguide cross sections.

Figure 4

(a) Circular cylindrical scatterer in free space, with permittivity ${ϵ}_1=12$, permeability ${\mu }_1=1$, and radius $r_1=0.0795\lambda$, where $\lambda$ is the free-space wavelength. (b) Radiation pattern for a line source located at point $P$ with polar coordinates $r_s=0.5r_1$ and ${\phi }_s=180\mathrm{deg}$. (c) The cylinder in (a) covered with a metamaterial shell with permittivity ${ϵ}_2=-3$, permeability ${\mu }_2=-4.5$, and radius $r_2=1.45r_1$, resulting in a scattering cross-section reduction from $0.64\lambda$ to $0.01\lambda$. (d) Radiation pattern for the same source as in (b), but with the metamaterial cover.